3   ABSOLUTELY CONVERGENT SERIES       101
condition, proceed as in (5.3.3). To show that the condition is necessary, use contradic-
tion: there would exist an a > 0 and an infinity of finite subsets Hfc (k = 1, 2, . . .) of N,
no two of which have common points, and such that || £ xn\\ 2* a for each k; starting
neHfc
from the existence of these subsets, define a for which the series (jca(n)) is not conver-
gent.)
 (b)	Suppose the series (xn) is such that, for any strictly increasing sequence («*) of
integers, the series (xn^) is convergent. Show that the series (xn) is commutatively con-
vergent (use the same argument as in (a)). Prove the converse when E is complete
(use the criterion proved in (a)).
 (c)	If E = Rn, show that any commutatively convergent series in E is absolutely con-
vergent (use Problem 3 and the criterion of (a)).
 (d)	Extend the associativity property (5.3.6) to commutatively convergent series.
5.	Let E be the real vector space consisting of all infinite sequences x = (£n)nzo of real
numbers, such that   lim £„ = 0. For any x e E, let \\x\\ = sup |fn|.
n-voo	n
 (a)	Show that hjc|| is a norm on E, and that E, with that norm, is a Banach space (the
" space (c0) " of Banach).
 (b)	Let em be the sequence (Smn)n>0 , with Smn = 0 if m ^ n,8mm — I . Show that for every
00
point jc = (£n)eE, the series *££nen is commutatively convergent in E, and that its
n = 0
sum is x; give examples in which the series is not absolutely convergent.
6.	(a)   Let ($„) an increasing sequence of numbers >0 which tends to + oo. Show that the
series of general term (sn — sn-.i)/sn has an infinite sum. For each number p > 0, show
that the series of general term (sn — sn,i)lsHs£-i is convergent; compare to the series of
general term
(b)   Let (un)n%Q be a sequence of numbers ^0 such that w0>0, and let sn = ^uk
Jt = 0
for each n $» 0. Show that a necessary and sufficient condition for the series of general
term un to be convergent is that the series of general term un/sn be convergent (use (a)).
7.	Let («„) be a convergent series of numbers ^0. Show that there exists an increasing
sequence (cn) of numbers >0, such that  limcn= +00, and that the series (cnun) is
H-+CO
convergent.
8.	Let (un) be a convergent series of numbers ^= 0. Show that
lim
n-»oo
and
/l(«H-l)
/        1      1     1 \
I write -«r	tt =	71 •
\         n(n+l)     n     n + l}